As I read the spell, the target must make a saving throw when you cast the spell.
If he passes the saving throw, he becomes staggered for one round.
If he fails the saving throw, nothing happpens but he starts making saving throws on his turns for the next 3 rounds. Each time he makes one of those three, he can avoid moving along the track to suffocation. If he does manage to make all three, it seems he can avoid all adverse effect from the spell (including the staggered condition he would have had if he had passed the first saving throw).
It does say that if the target fails the first saving throw, the air is expelled from his lungs and he starts to suffocate. There doesn't seem to be any "condition" which covers the situation but it would seem to me that this would prevent things like speaking, spell casting, breath weapons as well as give some sort of condition like staggered for the duration.
How would you rule the spell if the first saving throw was failed but then subsequent ones were passed?
...this would prevent things like speaking, spell casting, breath weapons...
Though I wouldn't carry over staggered, I would treat it as you do. That is, if you can't do it underwater (for lack of air) you can't do it while under the effects of this spell.
We had this spell come up in our game last night. We ran it as the first failed save was the first of 3, but the effects only started on the person's turn.